An ethylene/styrene (aromatic vinyl compound) copolymer is known (Patent Document 1). Such a copolymer shows characteristics as an elastomer and further, it shows mechanical properties similar to soft vinyl chloride resins and has functional properties such as oil resistance and scratch or abrasion resistance. Further, an ethylene/styrene copolymer is also known wherein the alternating structure of ethylene and styrene contained in the copolymer has an isotactic stereoregularity (Patent Documents 2 and 3). As compared with a copolymer having no such a stereoregularity, this copolymer has restrictive crystallinity (microcrystallinity) based on the alternating structure and thus has characteristics such that the mechanical properties are further improved, and functions such as heat resistance and oil resistance are improved.
However, the above ethylene/styrene copolymers are statistic copolymers (so-called random copolymers), of which the copolymerization type may be described by Bernoulli, primary or secondary Markov statistics and thus have had drawbacks such that they are inherently deficient in heat resistance and also deficient in compatibility with e.g. a styrene type polymer. Further, although their mechanical properties are more similar to soft vinyl chloride resins than an olefinic polymer such as LLDPE (linear low density polyethylene), mechanical properties more similar to soft vinyl chloride resins are desired.
Therefore, a method for producing a so-called cross copolymer has been proposed, which is a method of copolymerizing a small amount of divinylbenzene to the ethylene/styrene copolymer to introduce dissimilar polymer chains by anionic polymerization via vinyl groups of divinylbenzene units (Patent Document 4). By this method, a styrene monomer in the polymerization solution may simply be taken into the polymer at a high conversion ratio, thereby to present a polymerization method having a very high efficiency. Further, the polymer (cross copolymer) thereby obtainable has heat resistance improved over the ethylene/styrene copolymer. However, each of the cross copolymers disclosed in Examples, has a polyethylene crystallinity and crystalline melting point, whereby the softness is substantially lost as compared with the ethylene/styrene copolymer. Further, the transparency is also substantially deteriorated as compared with the ethylene/styrene copolymer, and it is substantially opaque.    Patent Document 1: Japanese Patent No. 2,623,070    Patent Document 2: JP-A-9-309925    Patent Document 3: JP-A-11-130808    Patent Document 4: JP-A1-00/037517